Linearization of the product of symmetric orthogonal polynomials (Q1337045)

Let \((p_ n)_{n\geq 0}\) be a system of monic symmetric orthogonal polynomials defined by its three-term recurrence relation. This paper investigates an associated partial difference equation in two dimensions with solutions of the form \(p_ n(x)p_ m(x)\). To this end, an auxiliary function in four variables is introduced which may be regarded as a discrete analogue of Riemann's function for hyperbolic partial differential equations. These functions are closely related to the linearization coefficients of the products \(p_ n p_ m\), and, in fact, the author's method leads to explicit representations of the product linearization coefficients for some examples. For instance, the associated Hermite polynomials, their continuous \(q\)-analogues, and the associated continuous \(q\)-ultraspherical polynomials are treated in this paper where the linearization coefficients are \({_ 3 F_ 2}\) and \({_ 3\Phi_ 2}\) functions, respectively.